2024/06/15 by Zhao, Bowen, Andersson, Lars, Yau, Shing-Tung
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2406.10751
We look at the strong field behavior of the Wang-Yau quasi-local energy. In particular, we examine the limit of the Wang-Yau quasi-local energy as the defining spacelike 2-surface Σ approaches an apparent horizon from outside. Assuming that coordinate functions of the isometric embedding are bounded in W2,1 and mean curvature vector of the image surface remains spacelike, we find that the limit falls in two exclusive cases: 1) If the horizon cannot be isometrically embedded into R3, the Wang-Yau quasi-local energy blows up as Σ approaches the horizon while the optimal embedding equation is not solvable for Σ near the horizon; 2) If the horizon can be isometrically embedded into R3, the optimal embedding equation is solvable up to the horizon with the unique solution at the horizon corresponding to isometric embedding into R3 and the Wang-Yau quasi-local mass admits a finite limit at the horizon. We discuss the implications of our results in the conclusion section.